Solving Dyson-Schwinger equations by chord diagrams Karen Yeats - - PowerPoint PPT Presentation

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Solving Dyson-Schwinger equations by chord diagrams

Karen Yeats

University of Waterloo

Algebraic and combinatorial perspectives in the mathematical sciences seminar June 4, 2020 Thanks to NSERC and the CRC program.

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Rooted chord diagrams

A rooted chord diagram with n chords is a matching on {1, 2, . . . , 2n}. We can draw it on a line or on a circle. The directed intersection graph of a chord diagram has a vertex for each chord, and an edge between chords {a, b}, {c, d} (a < b, c < d) if a < c < b < d, i.e. the chords cross.

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Connectivity and terminal chords

A rooted chord diagram is connected if the directed intersection graph is weakly connected. A chord is terminal if it has no outgoing edges in the directed intersection graph.

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Chord orders

There are two important ways to order the chords of a chord

• diagram. Both extend the partial order given by the directed

intersection graph. Order chords by left end-point. Order chords recursively as follows:

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The world of perturbative quantum field theory

A picture of perturbative quantum field theory:

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Feynman graphs and Feynman integrals

So from a combinatorial perspective, we have something like generating series for Feynman graphs weighted by the Feynman integrals. An example Feynman integral:

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A Dyson-Schwinger equation

Like how in enumerative combinatorics we get functional equations for the generating functions from decompositions of the objects, we get Dyson-Schwinger equations in quantum field theory. Take an example like G(x, L) = 1 − x q2

• d4k

k · q k2G(x, log(k2))(k + q)2 − same

• q2=µ2

which is a Dyson-Schwinger equation for a piece of Yukawa theory, solved by Broadhurst and Kreimer. Which piece?

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Rearranging

From G(x, L) = 1 − x q2

• d4k

k · q k2G(x, log(k2))(k + q)2 − same

• q2=µ2

We can expand 1/G(x, log(k2)) as a series in x and log(k2), use log a = ∂a|=0, recollect the series, to get G(x, L) = 1 − xG(x, ∂−ρ)−1(e−Lρ − 1)F(ρ)|ρ=0 where F(ρ) is the regularized integral for the one-loop diagram.

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. . . we find new diagrammatics

Interpret G(x, L) = 1 − xG(x, ∂−ρ)−1(e−Lρ − 1)F(ρ)|ρ=0 as an equation in formal power series. Recursively it determines the coefficients of G(x, L) in terms of F(ρ) = f0ρ−1 + f1 + f2ρ + · · · This works for the specific F(ρ) of the Yukawa example, or for general F(ρ), also for more general Dyson-Schwinger equations G(x, L) = 1 −

• k≥1

xkG(x, ∂−ρ)1−sk(e−Lρ − 1)Fk(ρ)|ρ=0 These also have expansions as sums over combinatorial objects – new diagrammatics.

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The initial result

with Marie Theorem G(x, L) = 1 − xG(x, ∂−ρ)−1(e−Lρ − 1)F(ρ)|ρ=0 with F(ρ) = f0ρ−1 + f1 + f2ρ + · · · is solved as a series by G(x, L) = 1 −

• C

x|C|ftk−tk−1 · · · ft2−t1f |C|−k

• i≤t1

ft1−i (−L)i i! where the sum is over rooted connected chord diagrams and the t1 < t2 < · · · < tk are the terminal chords of the chord diagram C. Generalization with Hihn.

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Questions this asks for combinatorics

Chord diagrams are longstanding objects investigated by

• combinatorialists. However

The terminal chords are a new parameter. What can we say about the statistics of the gaps between terminal chords? Some results with Courtiel. The recurrences that appear reminded N. Zeilberger of recurrences he saw in lambda calculus and led to a new bijection with bridgeless maps (with Courtiel and Zeilberger). Invariance under choices? Bijective connections between different classes of chord diagrams which arise? (ask Ali Mahmoud and Lukas Nabergall) Can we categorify? (ask Lucia Rotheray)

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Questions this asks for physics

Feynman diagrams give combinatorially indexed expansions, but each gives something complicated. This is an expansion in combinatorial objects where each object contributes something very simple. The chord diagram expansion is particularly nice for looking at the leading log expansion, next to leading log expansion etc. (with Courtiel) What does it tell us about which terms dominate the perturbative expansion? What does it tell us about the renormalization group equation? How generally does it apply (some work in progress by Lukas Nabergall)? Could it truly be an alternate diagrammatics for perturbative quantum field theory?

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Questions for everyone

But why? Why chord diagrams? Is there a natural way to go directly from Feynman diagrams to chord diagrams, at least in the Yukawa case. Kurusch and I are working on this. Can the proof of the chord diagram expansion be made more combinatorial – yes, work in progress by Lukas Nabergall.

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Next-tok leading log expansions

Let’s discuss one result in more detail, see arXiv:906.05139 (with Courtiel) First recall G(x, L) = 1 −

• C

x|C|ftk−tk−1 · · · ft2−t1f |C|−k

• i≤t1

ft1−i (−L)i i! Note, G(x, L) is triangular. The degree of L is at most the degree

• f x in each term.

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Leading log part; next-to-leading log part

The leading log part is The next-to-leading log part is

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After that

After that something different happens. We had G(x, L) = 1 −

• C

x|C|ftk−tk−1 · · · ft2−t1f |C|−k

• i≤t1

ft1−i (−L)i i!

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More general chord diagram expansions

But for the results of arXiv:906.05139 we need more general G(x, L). Now our chord diagrams have weighted chords. There is a parameter s indicating the insertion growth rate. They are ω-marked: the intervals covered by c contain d(c)s − 2 marks.

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Algorithms for the expansions

The analogous expansion in these objects solves the generalized Dyson-Schwinger equation (Hihn,Y rephrased by Courtiel). There is an automatable procedure to calculate any particular next-tok leading log expansion. Julien implemented it in Maple. It extends the ideas we saw in the simpler case – identify which families of chord diagrams contribute.

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Dichotomy

There is a dichotomy between s = 1 and s ≥ 2. Theorem (Courtiel) For s ≥ 2 The dominant types in the next-tok leading log expansion are those with k1 adjacent terminal chords of weight 1, first terminal chord of weight 1, k2 non-terminal chords of weight 2, and all other chords non-terminal of weight 1 (k1 + k2 = k). The number of these diagrams grows like (s − 1)k1 Γ(1 − 1

s )k!

k, k1 log

• (n)kn− 1

s −1sn−k−1.

The leading terms in the expansion involve only a1,0, a2,0 and a1,1.

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Theorem (continued) For s = 1 The dominant types in the next-tok leading log expansion are those with only one terminal chord of weight 2, k non-terminal chords of weight 2, and all other chords non-terminal of weight 1. The number of these diagrams grows like 1 (k − 1)! log(n)k−1n−2. The leading terms in the expansion involve only a1,0 and a2,0.

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Extra space